N5 Top-Up Prelim Revision Questions - Web Version

N5 Formula Sheet

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The Cosine Rule

Hint

The formula sheet gives both forms of the Cosine Rule.

Which form is best suited for finding the length of a missing side?

Pay attention to where angle \(A\) fits into the formula when calculating side \(a\).

Answer

\(32.45\) cm

Solution

Hint

The formula sheet gives both forms of the Cosine Rule.

Which form is best suited for finding the size of a missing angle?

Pay attention to where side \(a\) fits into the formula when calculating angle \(A\).

Answer

\(68.0^{\circ}\)

Solution

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Algebraic Fractions

Hint

Simplfying algebraic fractions should begin by factorising.

In an exam question, expect both the numerator and denominator to share the same factor

Answer

\(\dfrac{x-5}{x-3}\)

Solution

Hint

To add or subtract fractions, the denominators should be the same.

Answer

\(\dfrac{13x+7}{15}\)

Solution

Hint

The lowest common multiple of \(x+3\) and \(x-2\) is \((x+3)(x-2)\).

Start by rewriting each fraction with \((x+3)(x-2)\) as the denominator, adjusting the numerators appropriately.

Answer

\(\dfrac{7x+11}{(x+3)(x-2)}\)

Solution

Hint

The lowest common multiple of \(3x\) and \(x^2\) is \(3x^2\).

Answer

\(\dfrac{5x-6}{3x^2}\)

Solution

Hint

Approach algebraic fractions in a similar manner to numerical fractions.

What is the approach for divisions?

Then use cross-cancelling to simplify.

Answer

\(\dfrac{k}{6p}\)

Solution

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Trig Equations

Hint

First rearrange the equation to the form \(\sin{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Answer

\(x=53.1,126.9\)

Solution

Hint

First rearrange the equation to the form \(\tan{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Answer

\(x=77.5,257.5\)

Solution

Hint

First rearrange the equation to the form \(\cos{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Which two quadrants apply when \(\cos\) is negative?

Answer

\(x=104.5,255.5\)

Solution

Hint

First rearrange the equation to the form \(\sin{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Which two quadrants apply when \(\sin\) is negative?

Answer

\(x=221.8,318.2\)

Solution

Hint

First rearrange the equation to the form \(\tan{x^{\circ}}=\dots\)

Remember to use the \(ASTC\) diagram to find two solutions.

Answer

\(x=53.1,233.1\)

Solution

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Quadratics II

Hint

A number of approaches to completing the square question this are possible.

Check your notes and examples to see which method your teacher has taught.

Answer

\((x+6)^2-2\)

Solution

Hint

A number of approaches to completing the square question this are possible.

Check your notes and examples to see which method your teacher has taught.

Answer

\((x-4)^2+5\)

Solution

Hint

The turning point for \(y=(x-a)^2+b\) is \((a,b)\).

Answer

\((-5,3)\)

Solution

Hint

Start by writing the equation of the parabola using \((4,-3)\).

Compare this question to the previous one.

For part (c), any parabola with a turning point at \((a,b)\) has a vertical line (axis) of symmetry with the equation \(x=a\).

Answer

\(a=-4\), \(b=-3\) and axis of symmetry has equation \(x=4\).

Solution

Hint

To determine the nature of the roots, use the discriminant, \(b^2-4ac\).

Begin by stating the values of \(a\), \(b\) and \(c\).

Compare the result to zero to make a conclusion about the roots.

Pay close attention to the words which must be used in the answer.

Answer

\(44>0\) so real, distinct roots.

Solution

Hint

Note that each of the three possible conclusions must include reference to **real roots*.

Answer

\(-16<0\) so no real roots.

Solution

Hint

The quadratic formula is provided on the formula sheet.

Begin by stating the values of \(a\), \(b\) and \(c\).

Answer

\(x=0.9,x=-2.4\)

Solution

Hint

Take care to substitute negative values in brackets.

If you are asked to solve a quadratic equation in an exam and get a math error in your calculator, check your working carefully, expecially for any negative values.

Answer

\(x=1.4,x=0.4\)

Solution

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Vectors

Hint

Substitute and evaluate. Remember that vectors are not fractions, so there should be no concerns with the "bottom numbers" being different.

Answer

\(\begin{pmatrix}19\\0\end{pmatrix}\)

Solution

Hint

Take care with negatives.

Answer

\(\begin{pmatrix}-2\\\phantom{-}2\\-6\end{pmatrix}\)

Solution

Hint

Finding the magnitude of \(\mathbf{u}=\begin{pmatrix}x\\y\end{pmatrix}\) requires the formula:

\(|\mathbf{u}|=\sqrt{x^2+y^2}\)

Answer

\(10\)

Solution

Hint

Finding the magnitude of \(\mathbf{u}=\begin{pmatrix}x\\y\\z\end{pmatrix}\) requires the formula:

\(|\mathbf{u}|=\sqrt{x^2+y^2+z^2}\)

Answer

\(\sqrt{35}\)

Solution

Hint

For (a), start by expressing the journey from P to R as a point-to-point journey:

\(\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}\)

Then substitude the vectors required.

For (b), use the answer from part (a).

What is vector \(\overrightarrow{RS}\) equal to?

Answer

1. \(\mathbf{u-v}\)

2. \(\frac{1}{2}\mathbf{u-v}\)

Solution

Hint

Use the steps outlined in the previous question.

Answer

1. \(\mathbf{a-b}\)

2. \(\frac{1}{3}\mathbf{a}+\frac{2}{3}\mathbf{b}\)

Solution

Hint

For (a), think about the \(x\), \(y\) and \(z\) coordinates one-by-one.

Note that M is above the centre of the pyramid.

For (b), start by finding a vector to describe the journey from \(M\) to \(G\).

The length of a vector is given by its magnitude.

Answer

1. M\((1.5,1.5,5)\)

2. \(2.92\) units

Solution

Hint

Start by writing down each vector in component form.

Answer

\(\begin{pmatrix}5\\1\end{pmatrix}\)

Solution